# Shear strain stress relationship

### Shear Stress Strain

The symbol used for shear strain is g (gamma). As with axial stress and strain, a relationship exists between Shear Stress and Shear Strain. Instead of Young's . The resulting stress-strain curve or diagram gives a direct indication of the G ⇒ Shear Modulus - Slope of the initial linear portion of the shear stress-strain. The relationship between the stress (force per unit area) and strain that a particular material The ratio of the shearing force to the area A is called the shear stress. If the object is twisted through an angle q, then the shear strain is: s t r a i n.

**Elastic Constants and their Relationship - Lecture - 5**

A material is said to be elastic if it returns to its original, unloaded dimensions when load is removed. Within the elastic limits of materials i.

## Stress–strain curve

There will also be a strain in all directions at right angles to s. The final shape being shown by the dotted lines. It has been observed that for an elastic materials, the lateral strain is proportional to the longitudinal strain. The ratio of the lateral strain to longitudinal strain is known as the poison's ratio.

### Stress strain relationships in rocks -

Consider an element subjected to three mutually perpendicular tensile stresses sxsyand sz as shown in the figure below. In the absence of shear stresses on the faces of the elements let us say that sxsysz are in fact the principal stress. The resulting strain in the three directions would be the principal strains. We will have the following relation.

For Two dimensional strain: Hence the set of equation as described earlier reduces to Hence a strain can exist without a stress in that direction Hydrostatic stress: The term Hydrostatic stress is used to describe a state of tensile or compressive stress equal in all directions within or external to a body. So let us determine the expression for the volumetric strain. Consider a rectangle solid of sides x, y and z under the action of principal stresses s1s2s3 respectively.

Let a cuboid of material having initial sides of Length x, y and z. Volumetric strains in terms of principal stresses: Futher -ve sign in the expression occurs so as to keep the consistency of sign convention, because OM' moves clockwise with respect to OM it is considered to be negative strain.

## Stress strain relationships in rocks

The other relevant expressions are the following: Let us now define the plane strain condition Plane Strain: In xy plane three strain components may exist as can be seen from the following figures: More details on the influence of stresses and the stress tensor can be found in Jeager and Cook [2] and Nye. The stresses must be balanced so that there is no acceleration of the body.

Standard stress conditions Several standard stress conditions are either assumed for analysis or modeling, or applied in the laboratory: Hydrostatic stress - all confining stresses are equal Uniaxial stress - one stress applied along a single axis other stresses are zero or held constant during an experiment Biaxial stress - two nonequal stresses applied third direction is equal to one of the others Triaxial stress - while this really represents three independent principal stresses, it is commonly used to represent separate vertical and two equal horizontal stresses e.

In contrast, increasing the pore pressure Pp tends to increase rock volume. Pp counteracts the effects of Pc.

### Stress–strain curve - Wikipedia

Thus, rock properties are controlled largely by the difference between Pc and Pp, or the differential pressure Pd. A more exact form will account for the interaction of the fluid pressure with the pore space and minerals and result in an effective stress Pe law Deformation, strain and modulus Application of a single vertical stress is one typical experiment run to measure material mechanical properties Fig.

- Shear modulus

If this stress continues to increase, eventually the material will fail when the uniaxial compressive strength is reached see Rock failure relationships. For the rest of this topic, however, we will deal only with small deformations and stresses such that the rock remains in the elastic region.

Vertical strain Restricting ourselves to small deformations and stresses, several important material properties can be defined. Normalizing this deformation by the original length of the sample, L, gives the vertical strain